Guess 2/3 of the Average - The Beauty Contest Game Solved

The setup

A hundred people each pick a number from 0 to 100. The winner is whoever comes closest to two-thirds of the room's average. What do you write?

Short answer: Pure logic drives the answer to 0 - but against real people, the winning number sits in the low twenties, so the right play depends on who is in the room.

Iterated elimination

The average cannot exceed 100, so two-thirds of it cannot exceed 66.7 - any guess above that is dominated. But if nobody sensible guesses above 66.7, the target cannot exceed 44.4. Then 29.6, then 19.8, and the ladder only stops at 0.

With every player perfectly rational and knowing everyone else is too, the equilibrium is everyone writing 0.

Level-k: the game inside the game

Real rooms are not infinitely rational. A level-0 player guesses randomly (average 50). Level-1 best-responds to that with 33. Level-2 responds to a room of level-1s with 22. Classic runs of this experiment land the winning number right around there - most humans do one to three steps of reasoning, not infinitely many.

So the genuinely correct answer is conditional: 0 against theorists, roughly 20 to 25 against a normal crowd. Saying exactly that - and why - is the winning move.

Why interviews ask it

This is Keynes's beauty contest, and it is asked because markets ARE this game: prices track what average opinion expects average opinion to be. The candidate who prices the room instead of the textbook is the one who gets the follow-up questions.